biomechanics
DESCRIPTION
BIOMECHANICS. Infla tation of latex tube – materi a l paramet er identification. NONLINEARITIES. s. e. In continuum mechanics. Geometric al n onlinearity Large displacements Large deformation. Material nonlinearity N onlinear constitutive equation. Constraints-contact - PowerPoint PPT PresentationTRANSCRIPT
Inflatation of latex tube – material parameter identification
BIOMECHANICS
NONLINEARITIES In continuum mechanics
Geometrical nonlinearity Large displacements
Large deformation
Material nonlinearity Nonlinear constitutive equation
Constraints-contact Boundary conditions
E
nE 0
ij ijkl klc
nE 0
Inflation test
Load
- Transmural pressure
- Axial force
Goal
- Constitutive equation of material
ModellingTensors of deformation
Gradient of deformation F
X(X1,X2,X3,t) – reference configuration x(X1,X2,X3,t) – loaded state.
Displacement U(X,t)=x(X,t)-X.
iij
j
x X X X tF
X
, , ,
1 2 3 tx X,
FX
In terms of displacements x=X+U
iij ij
j
UF
X
ModellingTensors of deformation
Green–Lagrange tenzor
TE F F I 12 ij ki kj ijE F F 1
2ji k k
ijj i i j
UU U UE
X X X X
12
Example:
One-dimenional homogeneous deformationí x1=X1, x2=X2, x3=X3.
Displacement U : U1=x1-X1, U1= X1-X1.
x xE E F F
X X
21 11 1 1 111 11 112 2 2 2
1 1
1 1 1 1
ModellingTensors of deformation
Engineering deformation
Example:
One-dimenional homogeneous deformationí x1=X1, x2=X2, x3=X3.
Displacement U : U1=x1-X1, U1= X1-X1.
jiij
j i
UUX X
12
X XU UX X X
1 11 11 1 111 2 2 2
1 1 1
2 2 1 1 1
Hyperelastic material
ijij
W
- true stress
- engineering deformation
Ronald S. Rivlin (1915-2005)
Melvin Mooney (1893-1968)
W c I c I Jd
2
10 1 01 2
13 3 1
I1, I2 first and second invariant of deformation tensor deviator, J change of volume, i main stretches
Function – density of deformation energy W.
Hyperelastic material
1. adiabatic
2. incompressible
i i
i
Wp
, ,
1 2 3
i…stretchesp…Lagrange multiplicator (pressure)
ExperimentInflation test
Measured quantities
1. Outer radius ro
2. Length of tube l3. Axial force F4. Internal pressure pi
Experimental setup
1. Sample2. Flanges (3.)
4. Weights5. Tank6. Syringe – pressure generator7. Syringe – weight adjustment8. T-cock9. Valve10. Pressure transducer11. Scale12. Stand13. Camera
Model - deformations Cylindrical coordinate system
, , , ,X Z R x z r Stretches i
t
o r rO R R
22 z
lL
r
hH
tangential axial radial
t
z
r
rR
lL
hH
F
0 00 0
0 0 0 0
0 00 0
Deformation gradient F
Model - deformations
o i o iV v R R L r r l 2 2 2 2
Incompressibility constraint
vJ
Vdet F 2 1
t
z t z r
r
det F
2
2 2 2 2
0 0
0 0 1
0 0
t z r
rlhRLH
1
Ro…initial outer radius Ri…initial inner radius ro…actual outer ri…and inner radius
L…initial length l…actual length
Model-membrane
i zz zzi
pr Gz F h r p r G
h rh:
20 2 0
2 2
j tt ttj
prt F hdz p rdz
h: 0 2 2 0
zz
tt dzG
Balance of forces
Stress from loads (p,G)
o i o o o t ztt
p r r p r h r rp rp p
h h h h H
2 1 12 2 2 2
tt tt tt ttz z
zzo i o i o
G GG Gr h r r h R R H H R H
2 2 2 2 2 2
i ot
i o
r rR R
o i o iR R L r r l 2 2 2 2
Outer radius is measured, inner radius is calculated from the incompressibility constraint
Stress from Const. Equat. Stresses from deformation energy
tt tt
W
zz zz
W
t z r t z r t z z r r tW W W c c, , , , 2 2 2 2 2 2 2 2 21 2 3 1 23 3
Mooney–Rivlin model W
Using incompressibility W is expressed as function of t, r
t r zJ 1 rt z
1 t zW W ,
Material parameter identification
Regrese
MODtt t
t
W c c,
1 2
MODzz z
z
W c c,
1 2
Model prediction Experiment
MOD o t ztt
rp
H
12
MOD tt z
zzo
GH R H
2 2
Goal function – least squares
n
EXP MOD EXP MODtt tt zz zz
j j
Q
2 2
1
Qmin
n is number of measurement (measured points)
Material parameter identification
Linear regression
n
EXP MOD EXP MODtt tt zz zz
j j
Q c c,
2 2
1 21
0
Stationary point (minimum) [c1*,c2
*]:
Q c c
c
,
1 2
1
0 Q c c
c
,
1 2
2
0 Q([c1*,c2
*])=minQ
Experiment
1. Assembly of measuring instruments
2. Measure dimensions in reference configuration (Ro,Ri,L,H,m)
3. Adjust camera
4. Flood pipe
5. Several test cycles (preconditioning) without records
6. At least 3 measuring cycles, recorded
7. Disassembly and cleaning
Experiment Measuring cycle
At least 6 measuring points ([ro,l,p,G]-loads and corresponding dimensions)
Upper limit for load ~ 20 kPa
Close the valve in each measuring point!
It is not necessary to measure in the region where the model assumptions are viaolated (buckling at higher loads)